Abstarct: The present thesis is an attempt to study fuzzy planar graph and the problem of fuzzy planar graph coloring. In this regard, the problems related to definitive planar graph have been extended to fuzzy planar graphs. The first chapter explores the history of the subject and the basic definitions of fuzzy graph. In the second chapter a definition of fuzzy planar graph and its duality, fuzzy multigraphs, 0.67- fuzzy planar graphs and its duality and the extension of Euler’s formula to fuzzy graph are discussed. The third chapter deals with the uniformity of planar fuzzy graphs, the extension of kuratowski theorem to fuzzy graphs, union of fuzzy planar graphs and definitions of regular and Platonic fuzzy graphs.In chapter 4, different colorings of alphas-cut fuzzy graphs and some of its applications as well as some concepts of fuzzy graphs such as degree of a vertex, degree of a face, face strength baxsed on triangular fuzzy numbers are addressed. Moreover, the coloring number and the concept of fuzzy coloring of a fuzzy graph baxsed on triangular fuzzy numbers are extended and 4-colored fuzzy and the relation between fuzzy coloring and fuzzy graph are discussed.